Slightly subcritical hypercube percolation

We will present some recent results about bond percolation on the hypercube $$\{0,1\}^m$$ in the "slightly" subcritical phase, that is, just below the critical percolation scaling window. We estimate the size, diameter and mixing time of the largest components. A difficulty that arises only in the subcritical phase is that the cluster of the largest size does not attain the largest possible diameter. Therefore, we are able to analyze rather accurately the cluster of largest diameter, but not the cluster of largest size, leaving some interesting open problems.

Joint work with Tim Hulshof.